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International Journal of Differential Equations
Volume 2011 (2011), Article ID 453727, 18 pages
http://dx.doi.org/10.1155/2011/453727
Research Article

Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity

Department of Mathematics, University of Salerno, 84084 Fisciano, Italy

Received 4 May 2011; Accepted 8 September 2011

Academic Editor: Julio Rossi

Copyright © 2011 G. Galise and A. Vitolo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with fully nonlinear second-order equations assuming a superlinear growth in u with the aim to generalize previous existence and uniqueness results of viscosity solutions in the whole space without conditions at infinity. We also consider the solvability of the Dirichlet problem in bounded and unbounded domains and show a blow-up result.

1. Introduction and Statement of the Main Results

We are concerned with the well-posedness of the fully nonlinear second-order uniformly elliptic problem with no limitation on the growth of and no condition on the behaviour of at infinity.

We will assume the following standard structure condition, which implies the uniform ellipticity: for and any .

As regards the monotonicity in the variable , we ask something more than the usual monotonicity assumption (): for and .

We also suppose, as generally with the formulation (1.1), that

We collect the above assumptions in the structure condition noting that fits into our framework.

Hence, this paper is in the wake of Brezis [1], who proved the existence and uniqueness of distributional solutions for the semilinear equation with and of Esteban et al. [2] for the case of -viscosity solutions of the fully nonlinear second order uniformly elliptic equation with .

Our aim is to extend the result in different directions, including lower-order terms, allowing the dependence on and going below the exponent .

Throughout the paper and are positive constants such that , called ellipticity constants, and will play the role of a Lipschitz constant.

Also, is the exponent such that for the generalized maximum principle (GMP) holds true; see Escauriaza [3], Crandall and Swiech [4], and Koike and Swiech [5]: if with and is an -strong solution of the maximal equation then with and being a positive constant depending on .

Here and in the sequel by -strong solutions of the equation we intend the functions which are solutions almost everywhere with respect to the Lebesgue measure (almost everywhere) as well as classical solutions to be pointwise solutions.

We also call -strong (classical) subsolutions, respectively supersolutions, of the - (-) solutions of the inequalities , respectively , almost everywhere (everywhere).

For the definition of -viscosity and -viscosity solutions, which are our main concern, we refer to Section 2. Correspondingly subsolutions, respectively supersolutions, in the viscosity sense will be referred to as solutions of the equation , respectively .

We establish a first result in the case of independent of .

Theorem 1.1. Let be a continuous function satisfying the structure condition such that for all . If , there exists a unique -viscosity solution of equation

To consider a dependence on , we need to control the oscillations in the variable , and this also requires a uniform bound of the local -norms of .

Theorem 1.2. Let satisfy the structure condition . Suppose also that for all there exist a constant and a function such that and as , and . If , , and then (1.1) has a unique -viscosity solution.

As it can be seen in Section 4, the structure condition is sufficient by itself for the existence. The uniqueness, as shown in Section 5, relies on a result of Da Lio and Sirakov [6] which is fundamental in our proof for the comparison between two solutions and . By virtue of this result, conditions (A2.1) and (A2.2) imply that the difference satisfies a maximal equation with a constant first-order coefficient. In particular, (A2.1) and (A2.2) are, respectively, stronger than the continuity of in the -variable and than the local summability of . Later on we also refer to these conditions as to the assumption:

In order to deal with merely measurable in , we will suppose for every there exists such that for and . We put

Then, we say that satisfies -estimates at if for all there exists a solution of the Dirichlet problem such that for some .

Finally, let

In the case , by Caffarelli [7], if is sufficiently “small” in a sense that will be made precise below, then . Such result was generalized by Escauriaza to the range with introduced above.

As a consequence, the structure conditions can be used “pointwise” to compare with almost everywhere obtaining a maximal equation for the difference to which GMP is applicable.

By virtue of the results of Winter [8] (see also Swiech [9]), the argument can be generalized to the case of merely measurable in the variable provided is convex in the matrix-variable .

Theorem 1.3. Let be a function satisfying the structure condition almost everywhere such that one of the following assumptions blocks holds true:(A3) is continuous and has -estimates for each with some ;(A3)′ is measurable in for all and convex in . Let in the case (A3), in the case (A3)′. If for every , with , then (1.1) has a unique -strong solution , provided that .

By our assumptions, the -strong solution of Theorem 1.3 will be also the unique -viscosity solution.

Theorem 1.3 can be used for instance in the case of Bellman-type equations where is a semilinear second-order operator with bounded measurable coefficients such that for almost everywhere and every , provided the are uniformly continuous in with continuity modulus independent of and .

Some cases of Isaacs-type equations can be treated with Theorem 1.3, as for instance (see [10]), the minimum between concave and convex operators, which are realized as infimum and supremum, respectively, of two families of semilinear operators, indexed by and , with the above conditions.

Next, consider a regular domain . If is bounded, in the case of a continuous , condition is sufficient in order that the Dirichlet problem with continuous boundary conditions has a -viscosity solution by [11, Theorem  1.1]. If is merely measurable, we will use the stronger condition of Section 4 needed for the existence of -viscosity solution in [11, Theorem  4.1].

The technique of the existence part of Theorem 1.3 allows to generalize such results to any regular domain, even unbounded, of . For other results in unbounded domains we refer to [12], where the case is considered limiting the growth of .

Theorem 1.4. Let be a domain satisfying a uniform exterior cone condition, and let be measurable in for all such that the structure condition holds almost everywhere . Then, for the Dirichlet problem has an -viscosity solution for every and every .

Remark 1.5. The solution of Theorem 1.4 is unique in the cases of Theorems 1.1, 1.2 and 1.3, where the structural and the additional conditions are to be intended correspondingly to hold for instead of .

Finally, a monotonicity argument can instead be used when is bounded from below to obtain boundary blow-up -viscosity solutions.

Theorem 1.6. Let be a domain satisfying a uniform exterior cone condition, and suppose that at least one of the assumption blocks of Theorems 1.1, 1.2 and 1.3 holds true for . Then, for the Dirichlet problem has an -viscosity solution for every such that .

Remark 1.7. Generally the existence results for the BVP of Theorems 1.4 and 1.6 fail to hold if the domain is not sufficiently regular. In fact, assuming uniformly elliptic such that Labutin [13] showed that the origin is a removable singularity for the equation that is every -viscosity solution in the punctured ball can be continued to an -viscosity solution in .

The paper is organized as follows. In Section 2 we introduce the notations and recall the main features of viscosity solutions which will be used. Then in Section 3 we prove a locally uniform bound which is the basic tool to construct the solutions in unbounded domains. The proof of the existence results will be given in Section 4, while the issue of uniqueness and blow-up is dealt with in Section 5.

2. Preliminaries

We will consider functions where is a domain (open connected set) of and the space of real symmetric matrices. The identity matrix will be denoted by and the trace of with Tr, while is one of the equivalent norms of in . For we put if We say that is uniformly elliptic with ellipticity constants and if for all and , where are, respectively, the maximal and the minimal (Pucci) operators in the class of uniformly elliptic operators with ellipticity constants and .

For denote by and the gradient and the Hessian matrix of . We wish to discuss the solvability of equation under the assumptions (1.2), (1.3) and (1.4), and we refer to satisfying (2.4) for all as classical solutions of the equation in . If and the equation is satisfied almost everywhere in , we call it an -strong solution.

We are interested in the weaker notion of solution in the viscosity sense.

Firstly suppose that is continuous in and continuous in . The function is a -viscosity subsolution, respectively supersolution, of the equation in if respectively for each and such that has a local maximum, respectively minimum, in . A function that is both a subsolution and a supersolution in the above sense is a -viscosity solution.

Here and below we denote by the ball of radius centered at , for short if .

Remark 2.1. If is a classical subsolution (supersolution) of , then is a -viscosity subsolution (supersolution) of . Conversely, if is a -viscosity subsolution (supersolution), then is a classical subsolution (supersolution); see [14, Corollary  2.6].
In the sequel we will also use the fact that, if is a -viscosity subsolution, respectively supersolution, of and is a -viscosity subsolution, respectively supersolution, of in , then the function , respectively , is a -viscosity subsolution, respectively supersolution, of the equation in , where , respectively .

Lemma 2.2. Let be, respectively -viscosity subsolution and supersolution of the equations and in , and assume (1.2)-(1.3). If at least one between and is in , then the difference is a -viscosity subsolution of the maximal equation in .

Proof. Let us suppose, for instance, that . Let be a -function such that has a local maximum in , then is a test function for , and by structure conditions (1.2)-(1.3) we have as claimed.

When is merely measurable in , we assume that the structure condition holds for almost every . Note that, if is continuous in , then (1.2) implies the uniform ellipticity.

Let ; then is called an -viscosity subsolution, respectively supersolution, of the equation in if respectively for each and such that has a local maximum, respectively minimum, in . A function that is both a subsolution and a supersolution is an -viscosity solution.

It is important that the generalized maximum principle (GMP) for -strong solutions of the maximal equation (see (1.10) at the beginning of the Introduction) continues to hold for -viscosity subsolutions as ; see for instance [5, Theorem  3.2].

Note that -viscosity solutions are -viscosity solutions of because the space of test functions for -viscosity solutions is larger than .

Conversely, if , assuming that and are continuous, then -viscosity solutions are -viscosity solutions; see [15, Proposition  2.9].

Remark 2.3. Suppose that , as before. If is an -strong subsolution (supersolution) of , then is also an -viscosity subsolution (supersolution); see [15, Lemma  2.6].
Conversely, if is an -viscosity subsolution (supersolution), then is an -strong subsolution (supersolution); see [15, Corollary  3.7].
Also, Remark 2.1 for , respectively , continues to hold for -viscosity subsolutions, respectively supersolutions, .
Finally, assuming (1.2)-(1.3) almost everywhere in , we infer that Lemma 2.2 continues to hold for -viscosity solutions.

For an extensive treatment of viscosity solutions see [11, 1517].

In the existence results we need some regularity of the domain of . We say that satisfies an exterior cone condition if for every point there exists a finite right circular cone with vertex such that . A uniform exterior cone condition means that all the cones are congruent to a fixed cone . For the use of these conditions see, for instance, [18] about the theory and [11] in the viscosity setting.

3. Uniform Estimates

In this section, following [2], we introduce Osserman’s barrier function where and (recall that ).

Lemma 3.1. Suppose for almost everywhere that for all , where and .
If one takes then the function , defined in (3.1), is an -strong solution of the equation

Proof. By the assumptions, it is sufficient to show that Since , where , then and the result follows by choosing as claimed.

In what follows is the exponent such that GMP holds for (see Sections 1 and 2) and .

Lemma 3.2. Let be a domain of such that . Suppose that satisfies structure conditions almost everywhere . If is an -viscosity solution of the equation with , then for each one has where , and and are positive constants.
Here,

Proof. By (SC) we have for all . Thus, from Lemma 3.1 we deduce that is an -strong supersolution of the equation On the other hand is an -viscosity subsolution of the equation Hence, by Lemma 2.2 (see Remark 2.3) the function is an -viscosity solution of the equation in .
Let be such that as and ; then and on . Therefore, applying GMP (1.10), we get from which for all .

Proposition 3.3. Let , , and be as in Lemma 3.2. If is an -viscosity solution of the equation then for each one has with , and as defined in Lemma 3.2.

Proof. From Lemma 3.2 we already know that The assertion will be proved showing the same inequality for .
To this end firstly observe that the function satisfies the equation where which turns out to satisfy (SC).
Therefore, is an -viscosity solution of the equation
On the other hand, by virtue of Lemma 3.1, is an -strong supersolution of the equation
Hence, we finish the proof arguing as in Lemma 3.2 to obtain the estimate for all .

4. Proof of the Existence Results

In this section, using the structure condition , or the slightly stronger variant defined below, we construct an -viscosity solution of the equation in under the assumption that with , where is the exponent above which GMP holds true; see Sections 1 and 2.

By the relationship between -viscosity and -viscosity solutions and between -viscosity and -strong solutions (see Section 2) the existence part of each one of Theorems 1.1, 1.2, and 1.3 will follow at once from Proposition 4.1 in rather general assumptions.

We will suppose that for all there exists a function such that as and almost everywhere in for , observing that it is satisfied by default if we assume that is a continuous function of . Then, we put It is worth to recall that condition is equivalent to in the case that is continuous.

Proposition 4.1. Let be measurable in and satisfy the structure condition almost everywhere for all . If , then equation has an -viscosity solution in for any .

Proof. Consider a smooth approximation of such that for every bounded domain in .
By [11, Theorem  4.1, Remark  4.8] we can solve in the -viscosity sense any Dirichlet problem for the equation in the ball with continuous boundary condition.
Choose a solution for each . Using Proposition 3.3, for we have where and are positive constants as defined in Lemma 3.2.
By the structure condition we have almost everywhere for .
Therefore for we have in with .
By -estimates (see [14, Proposition  4.10] and [19, Theorem  2]) we deduce that for a positive constant independent of .
By a diagonal process, using Ascoli-Arzelà theorem we extract a subsequence such that uniformly on every bounded domain.
From the stability results for -viscosity solutions, see [15, Theorem  3.8], is a solution of the equation.

Proof of Theorem 1.4. In the case we proceed along the same lines of the proof of Proposition 4.1 constructing -viscosity solutions , , of the approximating Dirichlet problems for a sequence such that for all bounded subsets .
Here, according to the notations of Proposition 3.3, , while is a continuous extension to of ; see for instance [20, Section  1.2].
Let . Since satisfies in turn a uniform exterior cone property, the existence of such follows from the assumptions on and the already mentioned [11, Theorem  4.1].
Furthermore, by Proposition 3.3, for we get The argument of the proof of Lemma 3.2 leads to inequality and therefore are equibounded in .
As a consequence, by -estimates they are equi-Hölder continuous in every subset with . By [19, Theorem  2] for every and , and therefore are also equicontinuous in .
Thus, using a diagonal procedure as in the proof of Proposition 4.1 we find an -viscosity solution of the Dirichlet problem under consideration.

5. Uniqueness and Blow-Up

In this section we begin noticing that from Section 3 we get at once the following maximum principle.

Proposition 5.1. Let , , and let be a domain of .
Suppose for almost everywhere that for all and is an -viscosity solution of the equation
(M1)If , then in .(M2)If and on , then in .Analogously, suppose for almost everywhere that for all and is an -viscosity solution of the equation (m1)If , then in .(m2)If and on , then in .

Proof. Let and . Firstly consider the cases (M1) and (M2). Since satisfies (SC), we can apply Lemma 3.2. Letting in (3.8) with , we get as asserted.
The other cases (m1) and (m2) can be treated by means of (M1) and (M2) considering the function and the operator are as in the proof of Proposition 3.3.

The above implies that, assuming , the function is the unique viscosity solution of the problem in .

Concerning the uniqueness for the inhomogeneous equation in , since solutions are considered in the viscosity sense, we need additional assumptions in order to use “pointwise” the structure conditions and thus to use the above maximum principle.

Also, we cannot in general employ the usual comparison arguments for Dirichlet problems (see for instance [16, Section  3]) not having in principle boundary conditions or bounds at infinity.

Proof of Theorem 1.1 (uniqueness). Let and be solutions of the equation . Set . We claim that , so that in .
Suppose on the contrary that . Since is continuous, arguing as in [21] and observing that are in , we can use the structure condition (SC) to have in . Using the maximum principle of Proposition 5.1 (M2), we should have in , a contradiction which proves our claim.
Interchanging the role of and , we also get in , and we are done.

Proof of Theorem 1.2 (uniqueness). Here we observe that, if and , then Proposition 3.3 implies that is bounded. In fact, if and we consider balls centered at , choosing and in (3.17), we get which is finite and independent of , by (A2.2).
Thus, if and are solutions of the equation , by (SC) and (A2.1) we can use of [6, Proposition  2.1] so that the difference satisfies a maximal equation in for some positive constant depending on , and .
Therefore, we can conclude as in the proof of Theorem 1.1 (uniqueness).

Proof of Theorem 1.3 (uniqueness). In this case we observe that both the assumptions () and (A3) or (A3)′ of the Theorem imply that -viscosity solutions of are in (see [14, Theorem  7.1], [3, Theorem  1], and [8, Theorem  4.2]). Hence are -strong solutions (see Remark 2.3), and we can use the structure condition to get the maximal equation (5.7) for the difference in , from which we conclude again using the maximum principle of Proposition 5.1 as in the proof Theorems 1.1 and 1.2 (uniqueness).

We insist on observing that the uniqueness of the solution of the Dirichlet problem depends on the fact that the difference between two solutions is a solution of a homogeneous maximal equation, and then we can invoke the maximum principle of Proposition 5.1. The same method will be used to prove by monotonicity the existence of blow-up solutions.

Proof of Theorem 1.6. Following [2] we consider a nondecreasing sequence of such that for all compact set of . Then by Theorem 1.4 we solve the problem As in the proof of Proposition 4.1, by a diagonal process, using , respectively , we find an -viscosity solution of the equation To compare and , we use with the additional assumptions (A1) or (A2), respectively with (A3) or (A3)′, to get a maximal equation for . Since is nondecreasing, then satisfies the boundary value problem Hence, using the maximum principle of Proposition 5.1 we get , that is .
Therefore is nondecreasing, and for we have for all ; whence the assertion follows.

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